Engineering Analysis/Linear Independence and Basis
||Before reading this chapter, students should know how to take the transpose of a matrix, and the determinant of a matrix. Students should also know what the inverse of a matrix is, and how to calculate it. These topics are covered in Linear Algebra.|
A set of vectors are said to be linearly dependent on one another if any vector v from the set can be constructed from a linear combination of the other vectors in the set. Given the following linear equation:
The set of vectors V is linearly independent only if all the a coefficients are zero. If we combine the v vectors together into a single row vector:
And we combine all the a coefficients into a single column vector:
We have the following linear equation:
We can show that this equation can only be satisifed for , the matrix must be invertable:
Remember that for the matrix to be invertable, the determinate must be non-zero.
Non-Square Matrix V
If the matrix is not square, then the determinate can not be taken, and therefore the matrix is not invertable. To solve this problem, we can premultiply by the transpose matrix:
And then the square matrix must be invertable:
The rank of a matrix is the largest number of linearly independent rows or columns in the matrix.
To determine the Rank, typically the matrix is reduced to row-echelon form. From the reduced form, the number of non-zero rows, or the number of non-zero columns (whichever is smaller) is the rank of the matrix.
If we multiply two matrices A and B, and the result is C:
Then the rank of C is the minimum value between the ranks A and B:
A Span of a set of vectors V is the set of all vectors that can be created by a linear combination of the vectors.
A basis is a set of linearly-independent vectors that span the entire vector space.
If we have a vector , and V has basis vectors , by definition, we can write y in terms of a linear combination of the basis vectors:
If is invertable, the answer is apparent, but if is not invertable, then we can perform the following technique:
And we call the quantity the left-pseudoinverse of .
Change of Basis
Frequently, it is useful to change the basis vectors to a different set of vectors that span the set, but have different properties. If we have a space V, with basis vectors and a vector in V called x, we can use the new basis vectors to represent x:
If V is invertable, then the solution to this problem is simple.
If we have a set of basis vectors that are not orthogonal, we can use a process known as orthogonalization to produce a new set of basis vectors for the same space that are orthogonal:
- Find the new basis
- Such that
We can define the vectors as follows:
Notice that the vectors produced by this technique are orthogonal to each other, but they are not necessarily orthonormal. To make the w vectors orthonormal, you must divide each one by its norm:
A Reciprocal basis is a special type of basis that is related to the original basis. The reciprocal basis can be defined as: